Geometry & Topology

Chord arc properties for constant mean curvature disks

William Meeks and Giuseppe Tinaglia

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We prove a chord arc type bound for disks embedded in 3 with constant mean curvature that does not depend on the value of the mean curvature. This bound is inspired by and generalizes the weak chord arc bound of Colding and Minicozzi in Proposition 2.1 of Ann. of Math. 167 (2008) 211–243 for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamental tool for studying complete constant mean curvature surfaces embedded in 3 with finite topology or with positive injectivity radius.

Article information

Geom. Topol., Volume 22, Number 1 (2018), 305-322.

Received: 4 November 2015
Revised: 12 March 2017
Accepted: 9 April 2017
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 49Q05: Minimal surfaces [See also 53A10, 58E12] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

minimal surface constant mean curvature minimal lamination positive injectivity radius curvature estimates one-sided curvature estimate chord arc


Meeks, William; Tinaglia, Giuseppe. Chord arc properties for constant mean curvature disks. Geom. Topol. 22 (2018), no. 1, 305--322. doi:10.2140/gt.2018.22.305.

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